Abstract: Hydrological sciences cover a wide variety of water-driven processes at the Earth’s surface, above, and below it. Data assimilation techniques in hydrology have developed over the years along many quite independent paths, following not only different data availabilities but also a plethora of problem-specific model structures. Most hydrologic problems that are addressed through data assimilation, however, share some distinct peculiarities: scarce or indirect observation of most important state variables (soil moisture, river discharge, groundwater level, to name a few), incomplete or conceptual modelling, extreme spatial heterogeneity, and uncertainty of controlling physical parameters. On the other side, adoption of simplified and scale-specific models allows for substantial problem reduction that partially compensates these difficulties, opening the path to the assimilation of very indirect observations (e.g. from satellite remote sensing) and efficient model inversion for parameter estimation. This chapter illustrates the peculiarities of data assimilation for state estimation and model inversion in hydrology, with reference to a number of representative applications. Sequential ensemble filters and variational methods are recognised to be the most common choices in hydrologic data assimilation, and the motivations for these choices are also discussed, with several examples.

Abstract: We introduce direct and inverse problems, which describe dynamical processes causing change in the Earth system and its space environment. A well-posedness of the problems is defined in the sense of Hadamard and in the sense of Tikhonov, and it is linked to the existence, uniqueness, and stability of the problem solution. Some examples of ill- and well-posed problems are considered. Basic knowledge and approaches in data assimilation and solving inverse problems are discussed along with errors and uncertainties in data and model parameters as well as sensitivities of model results. Finally, we briefly review the book’s chapters which present state-of-the-art knowledge in data assimilation and geophysical inversions and applications in many disciplines of the Earth sciences: from the Earth’s core to the near-Earth environment.